Monthly Archives: February 2019

Cyclic Groups

I’m quite busy this week, and so I’ve basically accepted that I am going to fall behind the plan that I laid out in my proposal. I hope to be able to work more quickly in the coming weeks to make up.

I had a meeting on Monday, in which we discussed several exercises from the chapter on cyclic groups. Most of the exercises from this chapter were fairly straightforward once you figure out how to approach them. In other words, they require very few steps to solve, but finding the correct angle of attack can be a challenge. For example, an exercise that we had some trouble with is as follows:

The key here is to observe that the order of every element of a cyclic group divides the order of the group, so if we raise an element to the p^n – 1 power, we should get the identity of the group. Then, if we raise an element to the p^n power, we should get the original element back. If we call the group in question G, then, notationally,

Note that p^n=p*p^(n-1), so we can write

So we have written a as a pth power of an element of the group. Because a is just an arbitrary element of the group, it applies to every element of the group, and so we are done.

This exercise and its solution highlights how the solutions often take advantage of basic divisibility facts, which is why they tend to be fairly straightforward once you find the proper method, but difficult until you manage to find the proper method.

 

Subgroups

In the last week of January and the first several days of this week, I worked through the chapter on subgroups and began the chapter on cyclic groups. I found most of the exercises from the subgroups chapter to be fairly straightforward, except for a couple that we discussed in our last meeting. I’ve included the interesting one below.

Finding the orders of the elements (in this notation, |a| denotes the order of element a) was not a challenge—in fact, it’s really a mechanical process. The tricky part was finding the relationship between the orders, like the last part of the question prompts. The orders are as follows:

  • |6|=2, |2|=6, |6+2|=3
  • |3|=4, |8|=3, |3+8|=12
  • |5|=12, |4|=3, |5+4|=4

There are some patterns that stand out right away, but none are quite precise enough to be the relationship that the book is likely looking for. For example, note that in every set, the orders of two of the elements multiply to give the order of the third. The issue is that there is no pattern to which ones multiply together and which one is given by multiplication of the other two. Another pattern is that all of the orders divide the order of the group. But this isn’t a relationship between the orders of ab, and a+b, and is also fairly straightforward, and is covered in the next chapter on cyclic groups (which I’m working through right now).

We also looked for a relationship involving some sort of greatest common divisor involving 12, a, and b, but couldn’t find anything there. We ended the meeting thinking that the pattern perhaps was that |a+b| divides |a||b|in other words, that the order of the sum divides the product of the orders.

I did some more research after the meeting on my own, looking into whether or not, given the orders of a and b, we can find the order of ab (note here I just switched to multiplicative notation, but it’s really the same as a+b was in the integers mod 12, because I’m just omitting the group operation). It turns out that we can’t, in general. we can’t say anything about the order of the products. I can’t understand the proof, because it relies on ring theory and some basic linear algebra (involving groups of matrices), but maybe by the end of the study I will be able to tackle it… Regardless, the result I found is Theorem 1.64 in this document, which I encountered via this MSE post.